Saturday, July 25, 2015

Bet you never knew how easy it is to get off the grid. Just go to the edge and jump off.

But, seriously, there will be a gap in comics for a while. As well as a gap in the usual places where I post updates. I'll be away and I don't know what access I'll have nor what (computer) tools I'll have available. And no one else knows how I operate any of the websites to run them in my absence.

Wednesday, July 22, 2015

Once again, Real Life manifests itself in odd ways. Sometimes I work around it. Sometimes L post late or take a day or two off. Not this time.

A family trip (not a vacation) was going to take me off the grid (mostly) next week and the week after. However, other separate issues have come up that make the rest of this week unlikely as well. Where does that leave me?

Posting comics is out. I might be able to doodle on my iPad, but I won't be able to edit the pictures nor compose the text. Text posts are possible, but I don't know how much time I'll have to compose one that isn't rambling (like this post is). And with nothing on the blog, there will be little to post on Google+ or Facebook. (Maybe my personal Facebook page, but not the comic page.)

Basically, that leaves me with occasional tweets on Twitter.

If I don't post another comic before Friday, I'll see you in a few weeks.

Saturday, July 18, 2015

Today's Probability Question is a Real Beauty

1. If all outcomes are equally likely, what is the probability that Mr. Burke has his arm around Miss America 2015?

2. In your opinion, do you think Mr. Burke cares about the answer? Er, I mean ... explain why question 1 is not a reasonable question.

Obviously, in the real world, not all outcomes are equally likely to win the pageant, but if you were to, say, pull a photo of one contestant from a pile, or were to pick a sash at random, you could calculate the probability that the picture or sash belonged to Miss New York.

Also, in the real world, before this past Thursday, Mr. Burke would never believe he'd ever had a photo of himself with his arm around Miss New York (Miss Jamie Lynn Macchia). However, in the real world, if Mr. Burke should ever get such a picture, there is a 100% certainty that he's going to post it!!!

Friday, July 17, 2015

Boolean is a logical value equal to True or False and is used in logic (such as in computer programming).

The interesting thing about them was when I found out that False has a zero value, but True isn't equal to 1. In fact, True is simply "not zero", literally any value other than zero (or possibly "null", which is something else and I'm not going there right now).

That means that in situations where you might write "If x != 0 then"(Note: != mean "not equal to" in many computer languages"), you could just write "If x then" because any value other than 0 (which we didn't want) would evaluate as True! This was great! ... except that my professors condemned this as "clever coding" which was harder to read, decipher and maintain, so Don't Do It.

I'm not a programmer by trade any more, but from what I read, clever coding could be the standard, for all I know, because it executes faster, and it's more of a feature of languages like C than it was of PL/I or Pascal.

Final note: "Boolean" is named after George Boole, so it gets a capital "B", like "Venn diagram".

Oh, and this is a bullion cube joke, if you didn't get that. You don't use "Boolean cubes", except maybe in roleplaying games.

Wednesday, July 15, 2015

For anyone who doesn't know what a "surd" is, I was one of you just a few weeks ago. I began brushing up on the Algebra 2 curriculum, in case I need to teach it in the fall, and I came across a page dedicated to the topic Surds.

Surds are just another name for radical numbers, particularly those that are irrational so the radical sign cannot be removed.

I checked with my son (the recent contributor and university student), and he hadn't heard the term. And then, a few days ago, a colleague in England, whom I follow on Twitter, posted some problems. The first response was "Surds! I love that topic!"

I admitted that I'd only recently heard that word, and the commenter asked, "Then what do you call them?"

The original poster responded, "Radicals!", so at least that term is known on the other side of the Pond.

Tuesday, July 14, 2015

There are multiple ways to solve problems with circles, but some will be more straightforward than others, based on the information given.

If a circle has a tangent drawn to it, then that tangent is perpendicular to the radius it intersects. In other words, a right angle is created. Generally speaking, that right angle will probably signal the need to use the Pythagorean Theorem at some point in the problem.

On the other hand, if the tangent is accompanied by a secant line, then a second theorem can be plucked from our toolkit: the Tangent-Secant Theorem. If the tangent and secant intersect at a point outside the circle, then the square of the length of the tangent from the external point to the circle will be equal to the product of the portion of the secant outside the circle times the length of the entire segment.

Consider the problem below:

Which of the two theorems do we need to use?

The answer is: either one of them.

The circle has three radii drawn, but only one is labeled. Write the "6" next to the other two segments.

You can now solve for x using the right triangle with legs 6 and x, and with hypotenuse 10. Or you can solve for tangent with length x using the secant with a length of 16 and an external length of 4.

If you choose to work both of them out, you'll find the same answer.

Keeping that in mind, look at this next problem:

Now you will see that you can make a diameter from the given radius, and create a secant. This gives you a second option for some for x.

Tuesday, July 07, 2015

When I was first started creating comics to post to my blog (which actually started a year and a half before I scribbled those Co-Medians in MS-Paint), one of my earliest attempts at Punnery was this comic about Transcendental Numbers

One thing that I never answered in the blog -- because I really didn't use the comic as an opportunity for learning at the time -- was just what is a transcendental number? (I could make another joke right now, but I'm writing on a notepad to use at a later date.)

Transcendental numbers are a type of Irrational number, but they aren't the same. In fact, when I create that "What Kind of Numbers Are There?" Venn diagram on the first day of Algebra class, I could actually include a subset circle on the Irrational side. (But, I don't for reasons similar to why I don't have an Imaginary oval next to the Real one. It's not going to matter any time soon.)

There are a lot of Irrational numbers (as we discussed a few days ago in a different post), including square roots of non-perfect squares. Those numbers aren't transcendental. Which numbers "transcend" these ordinary irrationals?

The definition of a Transcendental number is any number that is NOT Algebraic. Don't you just love negative definitions? Anyway, an Algebraic number is one that can be the root of a polynomial equation (such as a quadratic equation) which has only rational coefficients. Since x2 - 2 = 0 has rational coefficients, its roots -- plus or minus radical two -- are Algebraic. But some numbers cannot be roots of these equations. Which ones?

As the cartoon suggests, pi and e are two examples. The third one, gamma, is the Euler–Mascheroni constant (also called Euler's constant), which is found in Number Theory, and which I freely admit I was completely unaware of before 2008 when I looked up transcendental numbers to make the comic (and earlier this evening when I wanted to write this entry).

The last is phi (pronounced "fee"), which I've mentioned before in the blog and in the comics, which is related to the Golden Ratio. This has the distinction of being incorrect, but I'm not updating it eight years after the fact. I read some information back then that some considered it to be one for whatever properties, so I used it because I had a comic to make. However, according to the definition I used above, it can't be transcendental because it is the root of x2 - x - 1 = 0. (What other definitions or properties are out there? That is left an exercise for the reader to ascertain, if you really want to.)

Finally, closing out the joke with a further pun, because it is "Transcendental Meditation", each of the characters (with their little legs are cross as they could go) are saying "Ommmm....", which sounds like "Ohm", which is a unit of Resistance, which uses the letter omega. What I actually knew at the time because I had just read a book about it the previous summer, was that omega was actually the symbol for Chaitlin's constant, which -- drumroll -- is irrational and transcendental.

Seriously, I really did know about this, even if I was, at the time, a little fuzzy on what a transcendental number was. But, at least, now you're not.

Sunday, July 05, 2015

Reviewing some old papers and I found the following Graphic Organizer. Those were big that year, after they were stressed in a PD. I tried to make things a little more visual, even if it was only a flowchart with boxes and lines. Better than fill-in-the-blank problems, right?

Oddly enough, my initial thought for this sheet was to NOT have the students place the solutions in the final column. I felt that the actual solution was secondary to getting the solution. Or should I say "knowing how to get the solution"? However, students do like answering problems, even if they leave out all the steps in between. The first weeks I need to break them of bad habits, and drill into them the idea that it's not about WHAT the answer is as much as HOW you arrived at that answer and WHY it's correct.

Algebra is moving beyond Arithmetic. That's why we'll give you a calculator (or let you use your own) to do the arithmetic for you.

Saturday, July 04, 2015

I was planning to blog this Fourth of July something about Independent variables, given the nature of the day, but something else came up.

I was reading Twitter, which I do way more often than is probably healthy, and I saw another image posted (or reposted) by Jo Morgan (@mathsjem), which was a simple puzzle with 7 circles, arranged in 2 columns of three and an extra circle in the middle, such that there were three lines of numbers. The center circle was filled in with a number and you had to fill in the blank circles with certain numbers so that everything totaled a given amount.

With a little modification, I produced the following image. It's a first draft, and already I see one teaching mistake. If this is the first time we're approaching problems with two variables, then it isn't likely that I've discussed the concept of ordered pairs before, so that will likely change to just pairs.

The two takeaways I would want to see from this: first, that the students could create an equation with x and y and a constant equaling another constant (with Standard form being an extension); second, will they notice that as one number gets larger, the other, by necessity, must get smaller.

You can also keep asking for different possibilities until they "run out" or someone thinks to use a negative number or a fraction. (At that point, the jig is up! You can quit the exercise, unless you want to circle the room one full time to get an answer from everyone.

Here is the image I posted on twitter in response to the first one. Again, the instructions need to be updated/corrected, but I like the exercise itself.

Friday, July 03, 2015

In Algebra, what is a conjugate? First, it's a noun, not a verb, and it's pronounced something like CON-juh-git, depending on your regional accent, but NOT as con-jyoo-GATE, like a big Language Arts scandal blasted across front pages of the tabloids.

The conjugate of a binomial, an algebraic expression with two terms, is a second binomial with the same terms but the sign between them has changed from plus to minus or minus to plus.

For example, 3x - 7 and 3x + 7 are conjugates.

What makes them interesting? One property of conjugates is to make things GO AWAY, and if there's one thing that Algebra students like is when things go away. And since I refuse to leave, this is the next best thing.

Add, Subtract, Multiply

If you add two conjugates, you double the first term and eliminate the second: (3x - 7) + (3x + 7) = 6x
If you subtract two conjugates, you elimated the first term and double the second: (3x - 7) - (3x + 7) = -14
-- keeping the sign of the term in the first binomial.

If you multiply them, something interesting happens:

You get a Difference of Squares. That is, the square of the first term minus the square of the second term. When you do the Distributive Property, you should get two more terms -- and don't you forget that! -- but in this case, those terms will cancel out! (-xy) + (xy) = 0.
(3x - 7)(3x + 7) = 9x2 + 21x - 21x - 49 = 9x2 - 49.

This can be useful not just for multiplying binomials, but for multiplying actual, honest-to-goodness, Real numbers, too!
Take, for example, (16) X (24). Not really easy to do in your head, but if you split the difference, you can see that it is the same as (20 - 4)(20 + 4).

Voila! The answer is 384! Wasn't that easy? Isn't this the greatest trick?

Nah, it's not. Just Kidding, really.

Just use a calculator. Seriously. No one really wants to square "bad" numbers in their head and then subtract them! But sometimes, it's kinda cool and you can impress your friends if you carefully pick your numbers!

Radicals!

BUT WAIT, THERE'S MORE!

Suppose you have a binomial where one of the terms is a radical number. Wouldn't you like that to go away, too? Well you can! Just multiply it by the conjugate.

That's nice that you're cool with it, but you can't leave it alone. Suppose you have two divided by (6 plus radical 7). If there's a radical in the denominator of a fraction, it has to go away. That's just the rule. We're going to "simplify" it by multiplying both the numerator and the denominator of the fraction the conjugate, like this:

Isn't that so much better? It is, isn't it? Worth it, right? Right?

Imaginary Numbers

The same way that conjugates work for radical numbers, they can work with imaginary numbers.

If you have 3 + 4i, for example, in the bottom of a fraction again, you can make it real by multiplying by the conjugate, 3 - 4i.

Using our rule from about (3 - 4i)(3 + 4i) = 9 + 16 = 25, which looks suspiciously like a part of a Pythagorean Theorem problem -- but that's for another night.

Years ago, I saw an F-16 Fighting Falcon over Coney Island. (I think the pilot was "Major Kerry" or "Major Kenny" ... the PA announcer repeated it numerous times, but, again, years ago.) It was the first time I really experienced sound distortion -- the plane simply was NOT in the place where the sound was coming from.

Simplifying the numbers a bit. The plane traveled about 1,000 feet per second. Sound travels about 1,100 feet per second. Assume the plane was 1,100 feet in the sky. (It should have been no lower than 1,000 feet, I believe). That means that it took about one second to hear the planes engines roar. In that same second, it moved 1,000 feet across the sky -- nearly a fifth of a mile!

So I may be looking above the Parachute Jump and the plane's out over Breezy Point!
Okay, so you don't follow the geography, but you get the point. It's not where you want to look. And when you are looking at it, ignore the sound because there isn't another one behind you.

... Or is there?

Happy Fourth of July! ... a day early.

EDIT: I "remembered" it as an F15, probably because of all the other "F" sounds (and maybe because of the PA system). However, I did a web search anyway -- and I saw an article about the F-15 vs. the F-16 Fighting Falcon. Okay then, I remembered correctly! Nope, I didn't. I didn't read the article carefully enough. The F-15 in the article was, in fact, the Eagle. The correction was made above.

Thursday, July 02, 2015

Moving on from yesterday's discussion about Rational Numbers, what about irrational numbers, numbers which cannot be written as a ratio of two integers?

Most whole numbers have square roots which are irrational numbers, but not everything with a radical is irrational. The square root of a perfect square is perfectly rational. So how can you tell if a number is a perfect square without a calculator?

One way is through prime factorization. (Remember those factor trees from a long time ago. C'mon, they were fun to do -- and you can do them again.... just not when you're typing in a blog. Then, they're kind of a pain, but I'll try.)

Take a number such as 60. It's prime factorization is 2 X 2 X 3 X 5, or 22 X 3 X 5.
If we were to square 60, we'd multiply 60 X 60, but we could also multiply 22 X 3 X 5 X 22 X 3 X 5.
That number (3600) would have a prime factorization of 24 X 32 X 52.

Notice what happened to the exponents. They've all doubled from 1 to 2 or 2 to 4. Every time you square a number, the exponents of its prime factors double. So if a number has been square, then all of the exponents of its prime factors will be even numbers because they are multiples of two.

Going back to our original number, is the square root of 60 a rational number?

It could only be a rational number if 60 were a perfect square, and it can only be a perfect square if all the exponents of its prime factors are even. However, the prime factorization is 22 X 3 X 5. Only one factor is even, so it is not a perfect square and the square root of 60 is irrational.

But wait! There's more!

As long as we've done the legwork, there is one more thing that we can do. Radicals that are irrational can be simplified. This is done by factoring out the largest perfect square. If we look back at the prime factorization, 22 X 3 X 5, we can see that there are two factors of 2.

So the square root of 60 is the same as (the square root of 22) X (the square root of 3 X 5). The square root of 22 is just 2.
That means that the square root of 60 is (2) times (the square root of 3 X 5), or 2(radical 15).

Wednesday, July 01, 2015

The definition of a rational number is, naturally, any number that can be represented as a ratio. That is to say, it can be written as a fraction.
Showing that a Whole number or an Integer is rational is trivial: just place the number in the numerator of a fraction with a denominator of 1.

But what about numbers which are written as decimals?

Every middle schooler remembers the joys of converting fractions to decimals (and to percents, but that's a post for a different day). But what about converting them back to fractions?

Terminating decimals are a cinch. Take a number like .375, which is properly pronounced three hundred seventy five thousandths. (It's usually pronounced "point three seven five" by lazy people, including some educators, including myself. Shame on me.) They can easily be turned into a fraction where the denominator is a power of 10.

There are three decimal places, so if you multiply the decimal by 103, you have the numerator 375 and 1000 as the denominator: 375 / 1000, which simplifies to 3/8, which is obviously a rational number.

How can we convert repeating decimals into fractions?

Repeating decimals are a little trickier. Sure, you may recognize 0.7777777... and know that it's 7/9, or even 0.36363636... is 4/11. But what a strange number like 0.378378378...?

By multiplying and subtracting, the repeating pattern of the decimal can be removed, leaving a multiple of x that will always be one less than a power of 10. Dividing by the coefficient transforms the value into a fraction (which may or may not be reducible).

This can be done for any repeating decimal. So any repeating decimal can be written as a fraction, i.e., a ratio of two integers and is, therefore, a rational number.

About Me

Mr. Burke is a high school math teacher in New York as well as a part-time writer, and a fan of science-fiction/fantasy books and films.
He started making his own math webcomic totally by accident as a way of amusing his students and trying to make them think just a little bit more.
Unless otherwise stated, all math cartoons and other images on this webpage are the creation and property of Mr. Chris Burke and cannot be reused without permission.
Thank you.